2013
(May)
MATHEMATICS
(Major)
Course: 402
(A: Linear Programming, B: Analysis – II)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
1. (a) How many basic assumptions are necessary for all linear programming models? 1
(b) Write the general linear programming problem with decision variables and constraints. 2
(c) A company makes 3 models of calculator – A, B and C at factory I and factory II. The company has orders for at least 6400 calculators of model A, 4000 calculators of model B and 4800 calculators of model C. At factory I, 50 calculators of model A, 50 of model B and 30 of model C are made everyday. It costs Rs. 12,000 and Rs. 15,000 each day to operate factory I and II respectively. Find the number of days each factory should operate to minimize the operating costs and still meet the demand. 3
(d) Solve graphically the following LPP: 4
Maximize
Subject to the constraints
2. (a) For what reasons, additional variables are to be addend to convert an LPP into standard form? 2
(b) Solve by simplex method: 5
Maximize
Subject to the constraints
(c) Solve by twophase method: 8
Maximize
Subject to the constraints
Or
Solve by BigM method:
Maximize
Subject to the constraints
3. (a) What happens in the dual if the variables in primal in unrestricted in sign? 1
(b) Fill up the blank: 1
“If the primal problem has an unbounded objective function, then the dual has ____ solution.”
(c) Write two rules for constructing the dual from the primal. 2
(d) Find the dual of the following primal LPP: 4
Minimize
Subject to the constraints
Unrestricted in sign.
Or
Prove that the dual of the dual linear programming problem is the primal.
4. (a) What is the necessary and sufficient condition for the existence of a feasible solution to the transportation problem? 1
(b) Mention two properties of a loop in a transportation problem. 2
(c) Obtain an optimal solution to the following transportation problem by Vogel’s method: 9
Supply
 
19

30

50

10

7
 
70

30

40

60

9
 
40

8

70

20

18
 
Demand

5

8

7

14

34

Or
What do you mean by balanced transportation problem? Find the initial basic feasible solution of the following problem with the help of least cost method:
Supply
 
1

2

1

4

30
 
3

3

2

1

50
 
4

2

5

9

20
 
Demand

20

40

30

10

B: ANALYSIS – II
(Multiple Integral)
(Marks: 35)
5. (a) When will the trigonometric series be a Fourier series? 1
(b) For a periodic function of period, show that 2
Where is any number?
(c) If the functionis periodic with periodon the interval, then find the Fourier series of. 3
(d) Obtain the Fourier series of the periodic functionwith perioddefined as 4
And hence deduce that
Or
If a functionis bounded, periodic with periodand integrable onand piecewise monotonic on, then prove that
Where,are Fourier coefficients of.
6. (a) Write the parametric representation of the curve 1
(b) If a functionis defined as
Then show that
3
(c) Evaluate
Over the region 4
Or
Evaluate
Over the region
Where
(d) State and prove Green’s theorem. 5
Or
Using Green’s theorem, compute the difference between the line integrals.
And
Whereandare respectively the straight lineand the parabolic are, joining the pointsand.
7. (a) Define a surface in 1
(b) State Gauss’ theorem. 1
(c) Evaluate 2
Whereis the outer side of the part of the sphere 4
Or
Find the volume of the solid bounded by the surfaceand the plane
(d) Using Stokes’ theorem, show that 5
Whereis the portion of the surface
Or
Using Gauss’ theorem, show that
Whereis the closed surface bounded by the coneand the planeand,,are direction cosines of the outward drawn normal of.